Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well

1.   Introduction

In the present paper, we investigate the existence and concentration of the solutions to a class of nonlinear Kirchhoff-type problem

where $a, b > 0$ are constants, $\lambda > 0$ is a real parameter, $1 < q < 2$, $f \in C(\mathbb{R}^3\times\mathbb{R}, \mathbb{R})$ and the potential $V$ satisfies the following conditions:

$(V_1) \; V(x)\in C(\mathbb{R}^3)$, $V(x)\geq0$ on $\mathbb{R}^3$, and there exists $V_0 > 0$ such that the set $\{V < V_0\} = \{x\in\mathbb{R}^3|V(x) < V_0\}$ is nonempty and

where $|\cdot|$ is the Lebesgue measure and $\mathfrak{C}$ is the best constant for the embedding of $\mathcal{D}^{1, 2}(\mathbb{R}^3)$ in $L^6(\mathbb{R}^3)$;

$(V_2) \; \Omega_* =$int$V^{-1}(0)$ is nonempty and has a smooth boundary with $\overline{\Omega}_* = V^{-1}(0)$;

$(V_3)$ there exists an open set $\Omega\subset\Omega_*$ such that $V(x)\equiv0$ for all $x\in \overline{\Omega}$.

These kinds of hypotheses were first put forward by Bartsch and Wang ^[1] in their research on the nonlinear Schrödinger equations, and it has attracted the attention of several researchers, e.g., see ^{[2,3,9,12,13,16,27]}. We note that the conditions $(V_1)$ and $(V_2)$ imply that $\lambda V$ represents a potential well with the bottom $V^{-1}(0)$, and that its depth is controlled by $\lambda$. In view of this, we can expect to find the solutions which are concentrated at the bottom of the potential $V$ as the depth goes to infinity.

In recent years, the elliptic problem

has been widely studied by many researchers, where $a, b > 0$, $\Omega\subset\mathbb{R}^3$ is a smooth bounded domain and $f\in C(\Omega\times\mathbb{R}, \mathbb{R})$. Problem (1.2) is often referred to as a nonlocal problem on account of the presence of the term $\Big(\int_\Omega|\nabla u|^2dx\Big)\Delta u$, we do not know

from $u_n\rightharpoonup u$ in $H^1_0(\Omega)$, which implies that (1.2) is no longer a pointwise identity. This phenomenon causes some mathematical difficulties, but at the same time, it makes the research of (1.2) particularly interesting. Moreover, problem (1.2) has a profound and interesting physical context, which is related to the stationary analogue of the equation

where $u$ denotes the displacement, $f(x, u)$ is the external force, $a$ is the initial tension and $b$ is related to the inherent characteristics of the string (such as the Young's modulus). This hyperbolic equation generalizes the following equation:

which is proposed by Kirchhoff in ^[4] as an extension of the classical D'Alembert's wave equations for free vibrations of elastic strings. This model takes into account the changes in length of the string produced by transverse vibrations. For more mathematical and physical background on the Kirchhoff-type problem, we refer the readers to ^[5,6] and the references therein.

More recently, many researchers have been devoted to investigations into the Kirchhoff-type problem defined in the whole space $\mathbb{R}^3$, i.e., the following problem:

where $V\in C(\mathbb{R}^3, \mathbb{R})$ is a potential function and $f \in C(\mathbb{R}^3\times\mathbb{R}, \mathbb{R})$. There have been a lot of studies on the solutions of (1.3) under many different assumptions. See, for example, in ^[7], Li et al. obtained a positive solution for (1.3) by using the cut-off technique and monotone method. In ^[8], Li and Ye proved the existence of ground state solutions in the case of $f(x, u) = |u|^{p-1}u$ and $2 < p\leq3$. Later Ye ^[10] obtained a positive high-energy solution with superlinear nonlinearities by using the Nehari-Pohozaev manifold. For more results about Kirchhoff-type problems, readers can see ^{[11,14,15,17]} and the references therein.

Very recently, Du et al. ^[18] considered the following Kirchhoff-type problem:

where $a, b, \lambda > 0$ are parameters and the potential $V$ satisfies the conditions $(V_1)$ and $(V_2)$. They showed that the existence and asymptotic behavior of ground state solutions when $f(x, u)$ behaved similar to $|u|^{p-2} u$ with $4 < p < 6$. In ^[19], the authors obtained the existence of nontrivial solutions for the case of $f(x, u) = |u|^{p-2} u$ with $4\leq p < 6$. After that, Zhang and Du ^[20] obtained the positive solutions for $b$ small and $\lambda$ large by combining the truncation technique and the parameter-dependent compactness lemma when $f(x, u) = |u|^{p-2} u$ and $2 < p < 4$. Furthermore, Sun and Wu ^[21] proved the existence of generate multiple solutions when $f(x, u)$ was replaced by $f(x)|u|^{p-2} u$. Zhou and Zhu ^[22] got the existence and asymptotic behavior of ground-state solutions to (1.4) with general convolution nonlinearity. In particular, Choudhuri ^[23] obtained the existence of infinitely many solutions to a $p$-Kirchhoff-type problem without the Ambrosetti-Rabinowitz condition.

Motivated by the works mentioned above, the purpose of this paper was to study the existence and concentration of solutions for problem (1.1) with a sublinear perturbation term and steep potential well, which is different from the papers above. In addition, as is well known, this type of problem is characterized by its lack of compactness due to the embedding fails, which prevents us from using the variational methods in a standard way. We will construct some inequalities in order to apply them to recover compactness as $\lambda$ is considered to be large enough.

Before stating our results we need to introduce some notations and conditions.

Throughout this paper, we denote by $|\cdot|_p$ the usual norm of the space $L^{p}(\mathbb{R}^3)$, $1\leq p\leq \infty$, $C_1, C_2, ...$ stand for different positive constants and $g^\pm = \sup\{\pm g, 0\}$. For any $x\in\mathbb{R}^3$ and $r > 0$, $B_r(x): = \{y\in\mathbb{R}^3:|y-x| < r\}$. Moreover, if we take a subsequence of a sequence $\{u_n\}$, we shall denote it again as $\{u_n\}$. We use $o(1)$ to denote any quantity which tends to zero when $n\rightarrow \infty$.

Considering that $F(x, s) = \int_0^sf(x, \tau)d\tau$, which is the primitive of $f$, we suppose the following hypotheses:

$(f_1) \; f \in C(\mathbb{R}^3\times\mathbb{R}, \mathbb{R})$, where $f(x, s)s > 0$ for all $s\neq0$ and $f(x, s) = 0$ for all $s\leq0$, and it satisfies

uniformly for $x\in \mathbb R^3$;

$(f_2)$ there exist $\gamma_1 > 0$ and $p\in (2, 6)$ such that

$(f_3)$ there exists $\gamma_2\in\Big(0, \frac{\mathcal{S}\min\{a, 1\}}{4\mathfrak{C}^2_2(\mathcal{S}+1)}\Big)$ such that

where $\mathcal{S}$ and $\mathfrak{C}_2$ are positive constants (see Remark 2.2).

Remark 1.1. Obviously, $(f_1)$ and $(f_2)$ imply that for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

for all $(x, s)\in\mathbb{R}^3\times\mathbb{R}$. An example of nonlinearity in $f(x, s)$ satisfying the hypotheses $(f_1)$–$(f_3)$ is given by

where for each $i = \{1, 2\}$, $h_i$ is some positive continuous bounded function.

The sublinear perturbation $g$ is given by the following condition:

$(g_1) \; g\in L^{\frac{2}{2-q}}(\mathbb R^3)$ and there exists an open set $\Omega_g\subset\Omega$ such that $g > 0$ on $\Omega_g$. In addition, there exists $\gamma_3 > 0$ independent of $\varepsilon$ such that $\gamma_3 > C_\varepsilon$ for $d(p, q): = \frac{(p-2)^{p-2}\;(2-q)^{2-q}}{(p-q)^{p-q}}$ and $K: = \frac{(\min\{a, 1\})^{p-q}}{2^{p-q}\; \Big[\gamma_3 \mathfrak{C}^p_p \Big(1+\frac{1}{\mathcal{S}}\Big)^{\frac{p}{2}} \Big]^{2-q}}$, with $g$ satisfying

where $C_\varepsilon$ and $\mathfrak{C}_p$ are given by Remarks $1.1$ and $2.2$ respectively.

Now we may state our main results as follows:

Theorem 1.2. Suppose that $(V_1)$–$(V_3)$, $(f_1)$–$(f_3)$ and $(g_1)$ hold. Then, there exist $\widehat{b} > 0$ and $\Lambda_* > 0$ such that $b\in(0, \widehat{b})$, problem (1.1) has at least two solutions for all $\lambda > \Lambda_*$.

On the concentration of nontrivial solutions we have the following result.

Theorem 1.3. Let $u^{(1)}_\lambda$ and $u^{(2)}_\lambda$ be two solutions of problem (1.1) given by Theorem $1.2$ and $\Omega = \Omega_*$. Then, $u^{(1)}_\lambda\rightarrow u^{(1)}_0$ and $u^{(2)}_\lambda\rightarrow u^{(2)}_0$ in $E_\lambda$ as $\lambda\rightarrow \infty$, where $u^{(1)}_0\neq u^{(2)}_0\in H_0^1({\Omega})$ are two nontrivial solutions of

The outline of this paper is as follows. In Section 2, some definitions and preliminary results are introduced. In Sections 3 and 4, we prove Theorems $1.2$ and $1.3$.

2.   Preliminaries

In this section, we will establish the variational framework for problem (1.1) and prove some useful lemmas. We recall the definition of the Hilbert space $H^1(\mathbb{R}^3)$ endowed with the standard scalar product and norm

$\mathfrak{C}$ denotes the best Sobolev constant

where $\mathcal{D}^{1, 2}(\mathbb{R}^3): = \{u\in L^6(\mathbb{R}^3): \nabla u\in L^2(\mathbb{R}^3)\}$ is the Sobolev space with the norm

Let

be equipped with the inner product and norm

For $\lambda > 0$, we also need the following inner product and norm

If $\lambda\geq1$, then $||u||\leq||u||_\lambda$. Set $E_\lambda = (E, ||u||_\lambda)$, we have the following results.

Lemma 2.1. Suppose that $V(x)$ satisfies $(V_1)$. Then, $E$ is continuously embedded in $H^1(\mathbb{R}^3)$.

Proof. From the condition $(V_1)$ and the Sobolev inequality, we have

which implies that

This show that

which implies that $E$ is continuously embedded in $H^1(\mathbb{R}^3)$. This ends the proof.

Remark 2.2. (i) Let

For all $\lambda\geq\Lambda$, by using the same conditions and techniques in $(2.1)$ and $(2.2)$, we obtain

(ii) The embedding ${E_\lambda}\hookrightarrow L^p(\mathbb{R}^3)$ is continuous for $p\in[2, 6]$, and ${E_\lambda}\hookrightarrow L^p_{loc}(\mathbb{R}^3)$ is compact for $p\in[2, 6)$, namely, for all $u\in E_\lambda$ and $\lambda\geq\Lambda$, there are constants $\mathfrak{C}_p$ such that

Considering problem (1.1), we have the energy functional $I_\lambda:E_\lambda\rightarrow \mathbb{R}$ given by

Moreover, for all $u, v\in E_\lambda$, it is easy to see that $I_\lambda\in C^1({E_\lambda}, \mathbb{R})$ and

Hence, if $u\in E_\lambda$ is a critical point of $I_\lambda$, then $u$ is a solution of problem (1.1).

Next, we give the following variant of the mountain pass theorem (see ^[24]) where we consider the Cerami condition. Let $X$ be a Banach space and $I \in C^1(X, \mathbb{R})$. We recall that a sequence $\{u_n\}\subset X$ is said to be a Cerami sequence (in short $(Ce)_c$ sequence) at the level $c\in \mathbb{R}$ if $I(u_n)\rightarrow c$ and $(1+||u_n||_X)||I'(u_n)||_{X^*}\rightarrow 0$, where $X^*$ denotes the dual space of $X$.

Lemma 2.3. Let $X$ be a real Banach space. Suppose that $I \in C^1(X, \mathbb{R})$, $I(0) = 0$ and

$(A_1)$ there exist $\alpha, \rho > 0$ such that $I(u)\geq\alpha$ provided $||u||_X = \rho$;

$(A_2)$ there exists $e\in X$ with $||e||_X > \rho$ such that $I(e) < 0.$

Define

where

Then, there exists a sequence $\{u_n\} \subset X$ such that

3.   Proof of Theorem 1.2

In the next lemma we check that $I_\lambda$ satisfies the mountain pass geometry introduced in Lemma $2.3$.

Lemma 3.1. Suppose that $(V_1)$, $(V_2)$, $(f_1)$ and $(f_2)$ are satisfied. In addition, there exists $\widehat{b} > 0$ such that $b\in(0, \widehat{b})$. Then, $I_\lambda$ satisfies Lemma $2.3$ of $(A_1)$ and $(A_2)$ for all $\lambda\geq\Lambda$.

Proof. We can use the condition $(g_1)$, $(1.5)$, $(2.3)$ and the Hölder inequality to obtain

Let $B = \frac{\mathfrak{C}^q_2\Big(1+\frac{1}{\mathcal{S}}\Big)^{\frac{q}{2}}|g^+|_{\frac{2}{2-q}}}{q}$ and $D = \gamma_3\mathfrak{C}^p_p\Big(1+\frac{1}{\mathcal{S}}\Big)^{\frac{p}{2}}$. Again from the condition $(g_1)$, we see that $\frac{\min\{a, 1\}}{2} > \Big(\frac{B^{p-2}D^{2-q}}{d(p, q)}\Big)^{\frac{1}{p-q}}$, then we can take $\varepsilon\in \Big(0, \frac{\mathcal{S}\min\{a, 1\}}{2\mathfrak{C}^2_2(\mathcal{S}+1)}-\frac{\mathcal{S}}{\mathfrak{C}^2_2(\mathcal{S}+1)} \Big(\frac{B^{p-2}D^{2-q}}{d(p, q)}\Big)^{\frac{1}{p-q}} \Big)$, and by Lemma $3.1$ in ^[25], it is easy to see that there is $t_P > 0$ such that, for $\rho: = t_P = ||u||_\lambda$,

where $\Psi(t) = t^2-P t^p- Q t^q, P, Q > 0$, which finishes the proof $(A_1)$.

In order to prove $(A_2)$, notice from the assumption $(f_1)$ that

So, for any $\varepsilon > 0$, there exists $\delta > 0$ such that $F(x, u) > \frac{u^2}{\varepsilon}$ for all $u > \delta$. Let $R_\varepsilon = \frac{\delta^2}{\varepsilon}$, then $F(x, u) > \frac{u^2}{\varepsilon}$$\\ -\frac{\delta^2}{\varepsilon}$. Next, let $0 < \varphi\in C^\infty_0(\mathbb{R}^3)$ be fixed, we have

This implies

since $\varepsilon$ is arbitrary, by $(3.1)$ we obtain

Thus, using Fatou's Lemma, we have that

where $I_{\lambda, 0}(u) = I_\lambda(u)$ for $b = 0$. Therefore, if $\lim_{t\rightarrow +\infty}I_{\lambda, 0}(t\varphi) = -\infty$, then there exists $e = t\varphi\in E_\lambda$ with $||e||_\lambda > \rho$ such that $I_{\lambda, 0}(e) < 0$. Since $\lim_{b\rightarrow 0^+}I_\lambda(e) = I_{\lambda, 0}(e)$, we see that there exists $\widehat{b} > 0$ such that $I_\lambda(e) < 0$ for all $b\in(0, \widehat{b})$. This ends the proof.

Let $I_\lambda(u)|_{H^1_0(\Omega)}$ be a restriction of $I_\lambda$ on $H^1_0(\Omega)$, that is

for all $u\in H^1_0(\Omega)$. Define

where

Indeed, it is easily seen that $\widetilde{c}$ is independent of $\lambda$. Furthermore, if the conditions $(f_1)$ and $(f_2)$ hold, then by the proofs of Lemma $3.1$, we can conclude that $I_\lambda|_{H^1_0(\Omega)}(u)$ satisfies the hypothesis of the mountain pass theorem as in Lemma $2.3$. Since $H^1_0(\Omega)\subset E_\lambda$ for all $\lambda > 0$, one has $0 < \alpha\leq c_\lambda\leq \widetilde{c}$ for all $\lambda\geq\Lambda$. Now, we can take $M > \widetilde{c}$. Thus

In view of Lemmas $2.3$ and $3.1$ there exists $\{u_n\}\subset E_\lambda$ such that

where $c_\lambda$ is given by $(3.3)$.

Lemma 3.2. Suppose that $(V_1)$, $(V_2)$ and $(f_1)$–$(f_3)$ are satisfied. Then, the sequence $\{u_n\}$ defined by $(3.4)$ is bounded in $E_\lambda$ for all $\lambda\geq\Lambda$.

Proof. By using the condition $(f_3)$ and $(2.3)$, we have

we can deduce that $\{u_n\}$ is bounded in $E_\lambda$ for $n$ large enough. This ends the proof.

We are now ready to give the following compactness conditions for $I_\lambda$.

Lemma 3.3. Suppose that $(V_1)$, $(V_2)$, $(g_1)$, $(f_1)$ and $(f_2)$ are satisfied. Then, there exist positive constants $\beta, \mathcal{C} > 0$ such that $I_\lambda$ satisfies the $(Ce)_c$ condition in $E_\lambda$ for all $\lambda\geq \max \Big\{\Lambda, \frac{8\mathcal{C}^2}{\beta V_0}\Big\}$.

Proof. Let $\{u_n\}$ be a $(Ce)_c$ sequence. By Lemma $3.2$, we see that, up to a subsequence, $\{u_n\}$ is bounded in $E_\lambda$. Passing to a subsequence again if necessary, we may assume that there exist $u \in E_\lambda$ and $A \in \mathbb{R}$ such that

and

Set $v_n = u_n-u$. By the condition $(g_1)$, $(3.5)$ and the Brezis-Lieb Lemma ^[26], one has

and

Define

Now, we claim that $\Phi'_\lambda(u) = 0$. Indeed, from $I'_\lambda(u_n)\rightarrow0$, we have

for any $v \in E_\lambda$, which implies that $\Phi'_\lambda(u) = 0$. Next we prove that $u_n\rightarrow u$ strongly in $E_\lambda$. Then, $\{v_n\}$ satisfies exactly one of the following conditions:

Suppose that the case $(B_1)$ holds. Then, there exists a constant $\beta > 0$ such that

In view of the weakly lower semi-continuity of the norm, we have

Since the sequence $\{u_n\}$ is bounded in $E_\lambda$, there exists a positive constant $\mathcal{C}$ (independent of $\lambda$) such that

Then, by $(3.11)$ and $(3.12)$, one has

Define

Then, we can take $\lambda\geq \frac{8\mathcal{C}^2}{\beta V_0}$, one gets

Applying $(3.13)$ and $(3.14)$ leads to

By using the condition $(V_1)$, $|F_R|\rightarrow 0$ as $R\rightarrow \infty$. Combining $(2.3)$ and the Hölder inequality, we get

which implies that

for any $p\in[2.6)$ as $R\rightarrow \infty$. From $R\rightarrow \infty$ and $v_n\rightarrow 0$ in $L^p_{loc}(\mathbb{R}^3)$ with $p\in[2, 6)$, we have

which contradicts $(3.10)$, where $B^c_R: = \{x\in \mathbb{R}^3: |x|\geq R \}$. Thus, if the case $(B_2)$ holds, by the Lions Lemma ^[26], $v_n\rightarrow 0$ in $L^p(\mathbb{R}^3)$ for any $p\in(2, 6)$. Then, using $(1.5)$ and the Lebesgue dominated convergence theorem gives

It follows from $(3.6)$–$(3.9)$ and $(3.15)$ that

which implies that $v_n\rightarrow 0$ strongly in $E_\lambda$. This ends the proof.

The proof of Theorem 1.2. Under the assumptions of $(V_1)$–$(V_3)$, $(f_1)$–$(f_3)$ and $(g_1)$, according to Lemmas $2.3$, $3.1$ and $3.2$, for each $b\in(0, \widehat{b})$, set

then for all $\lambda > \Lambda_*$, there exists the $(Ce)_{c_\lambda}$ sequence $\{u_n\}$ for $I_\lambda$ on $E_\lambda$. Then, by Lemma $3.3$ and $0 < c_\lambda\leq \widetilde{c} < M$, we can obtain that there exist a subsequence $\{u_n\}$ and $u^{(1)}_\lambda\in E_\lambda$ such that $u_n\rightarrow u^{(1)}_\lambda$ strongly in $E_\lambda$. Furthermore, $I_\lambda(u^{(1)}_\lambda) = c_\lambda\geq\alpha > 0$ and $u^{(1)}_\lambda$ is a nontrivial solution for problem (1.1).

The second solution of problem (1.1) will be constructed by using local minimization. Now, we show that there exists $\phi \in E_\lambda$ such that $I_\lambda(l \phi) < 0$ for all $l > 0$ small enough. Using the condition $(g_1)$ and $(3.2)$, take $\phi\in H^1_0(\Omega)$ with $\int_\Omega g(x)|\phi|^q dx > 0$, we have, for all $l > 0$ small enough,

It follows from Lemma $3.3$ and $(3.16)$ that the minimum of the (weakly lower semi-continuous) functional $I_\lambda$ on any closed ball in $E_\lambda$ with a center $0$ and radius $R < \rho$ satisfying $I_\lambda(u)\geq0$ for all $u\in E_\lambda$ with $||u||_\lambda = R$ is achieved in the corresponding open ball and thus yields a nontrivial solution $u^{(2)}_\lambda$ of problem (1.1) satisfying $I_\lambda(u^{(2)}_\lambda) < 0$ and $||u^{(2)}_\lambda||_\lambda < R$. In addition, $(3.16)$ implies that there exist $l_0 > 0$ and $\varrho < 0$ independent of $\lambda$ such that $I_\lambda(l_0 \phi) = \varrho$ and $||l_0 \phi||_\lambda < R$. Then, we can conclude that

The proof is finished.

4.   Proof of Theorem 1.3

In this section, we investigate the concentration of solutions for $\lambda\rightarrow \infty$.

Proof of Theorem 1.3. Let $u^{(i)}_\lambda, i = 1, 2$ be the nontrivial solutions of problem (1.1) obtained in Theorem $1.2$. For any sequence $\lambda_n\rightarrow \infty$, let $u^{(i)}_n: = u^{(i)}_{\lambda_n}$ be the critical points of $I_{\lambda_n}$, namely, $I'_{\lambda_n}(u^{(i)}_n) = 0$ and

Then, similar to the proof in Lemma $3.2$, we have

then, there exists constant $C_1 > 0$ independent of $\lambda_n$ such that

Hence $u^{(i)}_n$ is bounded in $E_\lambda$. Thus, we assume that $u^{(i)}_n\rightharpoonup u^{(i)}_0$ weakly in $E_\lambda$ and $u^{(i)}_n\rightarrow u^{(i)}_0$ strongly in $L^p_{loc}(\mathbb{R}^3)$ for $p\in[2, 6)$. Now, we show that $u^{(i)}_n\rightarrow u^{(i)}_0$ strongly in $L^p(\mathbb{R}^3)$ for $p\in[2, 6)$.

Recall the definition of $A_R$ and $F_R$ in Lemma $3.3$, and that $|F_R|\rightarrow 0$ as $R\rightarrow \infty$ by the condition $(V_1)$. Then, for $\lambda_n \rightarrow \infty$, one has

Thus, by the Hölder and Sobolev inequalities, $(2.3)$, $(4.2)$ and $(4.3)$, we obtain

as $\lambda_n \rightarrow \infty$. Then, we infer that

as $R \rightarrow \infty$. Since $u^{(i)}_n\rightarrow u^{(i)}_0$ strongly in $L^p_{loc}(\mathbb{R}^3)$ for $p\in[2, 6)$, we have

Therefore, $u^{(i)}_n\rightarrow u^{(i)}_0$ strongly in $L^p(\mathbb{R}^3)$ for $p\in[2, 6)$. Set $w^{(i)}_n: = u^{(i)}_n-u^{(i)}_0$, use a similar argument to the proof in Lemma $3.3$, we can claim that $w^{(i)}_n \rightarrow 0$ strongly in $E_\lambda$.

Thus, by $(4.2)$ and Fatou's lemma, we have

Hence, $u^{(i)}_0 = 0$ a.e. in $\mathbb{R}^3\setminus\Omega$ and $u^{(i)}_0\in H^1_0(\Omega)$ by the condition $(V_2)$. Given $u^{(i)}_0\in H^1_0(\Omega)$, we obtain

for any $v^{(i)}\in H^1_0(\Omega)$. Finally, it follows from $(4.1)$ that

and

which implies that $u^{(i)}_0\neq 0$ and $u^{(1)}_0\neq u^{(2)}_0$. This completes the proof.

5.   Conclusions

In this paper, two nontrivial solutions are obtained for a Kirchhoff-type problem by using variational methods. Furthermore, the concentration behavior of solutions as $\lambda\rightarrow \infty$ is also explored. The results obtained in this paper are slightly different from previous works ^{[7,8,10,18,19,20,21,22]}. They may not have considered the existence and concentration of the solutions for Kirchhoff-type problems with sublinear perturbation and steep potential well. Therefore, the results of this paper expand the previous work to a certain extent.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11461058), the Natural Science Foundation of Sichuan Minzu College (Nos. XYZB2010ZB, XYZB2002ZA) and the fourth academic innovation team of Sichuan Minzu College (differential equation and dynamic system research team). The author would like to express their gratitude to the referees for the valuable comments and suggestions.

Conflict of interest

The authors declare no conflict of interest.